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*Nonlinear* methods are best applied when modeling and analysis procedures consider the inelastic material behavior of a structure. Nonlinear methods include static-pushover analysis and dynamic time-history analysis. If only elastic material behavior is to be considered, linear methods should be suitable. Linear methods include static strength-based analysis and dynamic response-spectrum analysis.

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Engineers may use any of these four analysis methods, shown below in Figure 1, to suit the following purposes:

* To characterize and gain insight into structural behavior; and

* To generate information useful for the design decision-making process.

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{center-text}Figure 1 - Analysis methods{center-text}

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Each analysis method has its benefits and limitations. An overview of each is described below:

* *Strength-based* analysis is a static-linear procedure where structural components are selected such that their capacity exceeds demand from loading. Strength-based demand-capacity (D-C) ratio indicate the adequacy of each component, and the need for redesign, if necessary. Modeling and analysis is fairly simple and straight-forward in that only the elastic stiffness properties are considered. Strength-based analysis is also the least time consuming.

* *Response-spectrum* analysis is a dynamic-linear method which plots maximum structural response as a function of structural period and damping. Response may be the acceleration, velocity, or displacement resulting from a given time-history record. In that response-spectrum analysis is a linear application, the superposition of gravity and lateral effects does not apply. As a result, structures must remain essentially elastic during response-spectrum analysis.

* *Static-pushover* analysis is a static-nonlinear procedure which indicates the elastic and inelastic structural response of a component or system subjected to monotonic loading which continually increases through an ultimate condition. This method produces a nonlinear force-deformation (F-D) relationship which provides insight into the ductility and limit-state behavior of a structure.

* *Time-history* analysis is a dynamic-nonlinear method which characterizes the dynamic response and inelastic behavior of a structure subjected to the time-history acceleration record of a ground motion (earthquake). The nonlinear material properties of ductile components, designed to yield under substantial loading, are modeled such that a step-by-step integration procedure may capture inelastic effect. Simultaneously, P-delta effect take into account the influence of gravity loading on deformed structural configuration. Output may provide data, plots, and animations describing structural response and component behavior.


h1. Capacity Design

During nonlinear analysis, ductility contributes an additional level of complexity to modeling and design. Ductile elements, either designed to specifically or possibly achieve inelastic behavior under certain loading scenario, may satisfy demand-capacity criteria through either of two parameters: strength or deformation.

It is best to select which elements will be permitted to yield (ductile components) and which will remain elastic (brittle components) during preliminary design. This way the designer determines how a structure will perform, and not the computational tools and analysis procedures. This is critical in that many sources of uncertainty are inherent to analytical modeling. A computer model may indicate behavior which will not actually occur in a real structure. Analysis models are only a simulation of physical phenomena that is impossible to predict exactly. Therefore it is best to select which elements will be permitted to yield, and implement those ductile systems from the beginning. This allows the engineer to create a more reliable model, and should lead to better structural design.

This frames the basic tenants of Capacity Design; structures should be designed with ductile systems predetermined. This way, systems permitted to yield may be designed with sufficient deformation capacity, and systems chosen to remain elastic are designed with sufficient strength capacity. To reiterate this point:

* Ductile components predetermined to undergo yielding shall be designed with sufficient deformation capacity such that they satisfy displacement-based demand-capacity ratio; and

* Brittle components which will remain elastic are designed to achieve sufficient strength such that they satisfy strength-based demand-capacity ratio.

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A benefit to Capacity Design is that only the elastic material properties of brittle components need to be modeled. If demand on any brittle component is found to exceed its capacity, redesign involves simply selecting an element with greater strength such that it remains elastic. Modeling only the linear projection of initial stiffness for all elastic elements simplifies analysis and reduces computational time and quantity of output data.


h1. Sources of nonlinearity

There are two types of nonlinearity, including geometric and material, described as follows:


h2. 1. Geometric nonlinearity

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Geometric nonlinearity, also known as P-delta effect, becomes critical when gravity load acts on a structural system which has displaced laterally. Secondary moments are then induced which contribute significantly to structural behavior. It is important to account for P-delta effect during nonlinear analysis. It doesn't increase computational time very much, and is accurate for drift levels up to 10%.

Large deformation analysis is another P-delta effect, except that it is denoted by the upper-case delta symbol. This type of geometric nonlinearity is assessed on the local level where equilibrium about column deformation is considered. This however only becomes significant in especially slender columns or at unreasonably large displacements. Further, it increases computational time significantly. An easier way to model this type of P-delta effect is to model additional nodes along a column length, which transfers the behavior (large displacement effect) into P-triangle.


h2. 2. Material nonlinearity

Material nonlinearity is the characterization of inelastic behavior within a component or system. Material nonlinearity is characterized by a force-deformation relationship. Two basic relationship types exist, including monotonic and hysteretic.


h5. Monotonic curve

Static-pushover analysis utilizes monotonic loading. Here, a component or system is subjected to a load condition where the independent variable, a deformation parameter, increases from zero to an ultimate value. The corresponding dependent variable, a force-based parameter, is plotted against deformation to produce a nonlinear, monotonic curve. Some examples of monotonic force-deformation relationships (and their mechanism) may include stress-strain (axial), moment-curvature (flexure), plastic-hinging (rotation), etc.

Figure 2 presents a static-pushover curve. Under monotonic loading, the force-deformation relationship begins linear-elastically, following the initial-stiffness tangent to a yield point. Inelastic behavior then begins, advancing through a series of limit states until an ultimate condition is achieved. Any strength-gain represents hardening, and strength-loss represents softening. After softening, a residual value may be achieved, which may sustain through unrealistically large displacements before reaching an ultimate condition. This nonlinear force-deformation relationship may then be simplified with little compromise to analysis accuracy through idealization as a series of linear segments, as shown in Figure 3. Please notice that immediate-occupancy (IO), life-safety (LS), and collapse-prevention (CP) limit states are denoted on the curve. While these parameters relate to structural serviceability, limit states may also be specific to plastic thresholds, as shown in Figure 4:

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{center-text}Figure 2 - Monotonic backbone curve{center-text}

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{center-text}Figure 3 - Idealized monotonic backbone curve{center-text}

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{center-text}Figure 4 - Serviceability limit states{center-text}


h5. Hysteretic cycle

Dynamic time-history analysis tracks the hysteretic behavior of a component or system subjected to cyclic loading. Here, material nonlinearity is plotted in a series of hysteresis loops. Rather than following a single monotonic curve to an ultimate condition, hysteresis repeatedly reverses the orientation of loading. Once some degree of inelasticity is achieved, behavior will begin to deviate from that of the monotonic curve with each unloading and reloading in the opposite direction. As shown in Figure 5, both stiffness and strength will deviate from their initial relationships as hysteretic cycles progress. Stiffness typically degrades, which is indicated by a decrease in slope upon load reversal. Strength levels may increase initially, but typically also degrade with cyclic behavior. A ductile system succeeds in maintaining its post-peak strength through hysteretic behavior and increasing levels of deformation.

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{center-text}Figure 5 - Hysteresis loop{center-text}

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Characterizing the development of strength and stiffness relationships, as they progress through dynamic time-history analysis, is an important feature of accurate nonlinear modeling. PERFORM-3D is a computational tool which provides this capability.

Depending on structural configuration and material, a hysteretic cycle may be one of many different types. Figures 6-10 illustrate some of the possible behaviors.

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h1. Conclusion

While accurate prediction of structural behavior is desirable, analysis models can only idealize the performance of real structures. Those using software tools should note that exact prediction of behavior is not possible. The objective of structural analysis is to generate information useful to the design decision-making process. Nonlinear methods enable greater insight into dynamic and inelastic structural behavior.